ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers Page 2 of 11


Complex Numbers - Representation
We construct complex numbers around the basic building block
j = –1 .
Note that most math books will use the letter i for this role, but in electrical engineering
we prefer to use the letter j since i is typically used to represent current.

Cartesian Representation of Complex Numbers
In its Cartesian form, a complex number is written in terms of a real part and an imagi-
nary part.

Example: z = 2 + 4j Im  z 
Re  2 + 4j  = 2 (Real part)
4 *
Im  2 + 4j  = 4 (Imaginary part)
(Note: the j is not included in the imaginary part.
The imaginary part is the number multiplying the j ).
2 Re  z 

We visualize complex numbers as points in a 2-D plane
where the x-axis is the real part and the y-axis is the imag-
inary part.


Copyright by Scott L. Miller Texas A&M University November 15, 2010
ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers Page 3 of 11


Complex Numbers - Representation
Polar Representation of Complex Numbers
In its polar form, a complex number is written in terms of Im  z 
magnitude, r , and phase,  . The polar form of a complex
4 *
number stems from Euler’s Identity,
r
e j = cos    + j sin    . 
Multiplying both sides by a magnitude, r , produces
2 Re  z 
re j = r cos    + jr sin    .

Therefore, for a complex number with magnitude, r , and
phase,  , we write the complex number in its Polar form as
z = re j and the Cartesian representation of the same com-
plex number would be
x = Re  re j  = r cos    , y = Im  re j  = r sin    .

The above equations show us how to convert from polar form to Cartesian form. The con-
version from Cartesian to Polar works out to be
y
r = x 2 + y 2 ,  = tan – 1  -- .
 x


Copyright by Scott L. Miller Texas A&M University November 15, 2010
ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers Page 4 of 11


Complex Numbers - Representation
Examples: Convert each of the following complex numbers from polar to Cartesian form.
(a) z = 2e j    4  (b) z = 3e –j (c) z = e j3  2

Im  z  Im  z  Im  z 

*
2 4
–3
Re  z 
* Re  z  Re  z 

 *




  3 3
2e j    4  = 2 cos  --- + j2 sin  --- 3e – j = 3 cos  –   + j3 sin  –   e j3  2 = cos  ------ + j sin  ------
 4  4  2  2

= 2+j 2 = – 3 + j0 = – 3 = 0 + j  –1  = –j




Copyright by Scott L. Miller Texas A&M University November 15, 2010
ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers Page 5 of 11


Complex Numbers - Representation
Examples: Convert each of the following complex numbers from Cartesian to polar form.
(a) z = 1 + j (b) z = 2j (c) z = – 1 – 2j
Im  z 
(a) r = x2 + y2 = 12 + 12 = 2
*
y 1 
 = tan – 1  -- = tan –1  --- = ---
 x  1 4 Re  z 

z = 1+j = 2e j    4 


(b) r = x2 + y2 = 02 + 22 = 2 Im  z 
*
y 2 
 = tan – 1  -- = tan –1  --- = ---
 x  0 2 Re  z 
z = 2j = 2e j    2 


(a) r = x2 + y2 =  –1 2 +  –2 2 = 5 Im  z 

y –2
 = tan – 1  -- = tan –1  ------ = 4.25rad
 x  – 1
Re  z 
z = – 1 – 2j = 5e j  4.25 
*



Copyright by Scott L. Miller Texas A&M University November 15, 2010
ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers Page 6 of 11


Complex Numbers - Arithmetic
Addition and Subtraction
It is easiest to add and subtract complex numbers in their Carte- Im  z 
sian from.
z 1 + z 2 =  x 1 + jy 1  +  x 2 + jy 2  =  x 1 + x 2  + j  y 1 + y 2  .
z1
z 1 – z 2 =  x 1 + jy 1  –  x 2 + jy 2  =  x 1 – x 2  + j  y 1 – y 2  . z2

z1 + z2 Re  z 
Visually, this works just like vector addition/subtraction. If you
are given complex numbers in polar form and you need to add or
subtract them, first convert them to Cartesian form, then add/subtract and then convert the
result back to polar form (if desired).

Multiplication and Division
It is easiest to multiply/divide complex numbers in their polar form
z 1 z 2 =  r 1 exp  j 1    r 2 exp  j 2   = r 1 r 2 exp  j   1 +  2   .
z1 r 1 exp  j 1  r
- = ----1 exp  j   1 –  2   .
---- = --------------------------
z2 r 2 exp  j 2  r2

There is no visual interpretation of multiplication/division.

Copyright by Scott L. Miller Texas A&M University November 15, 2010
ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers Page 7 of 11


Complex Numbers - Arithmetic
Multiplication and Division
You can multiply and divide complex numbers in the Cartesian domain, but it is a little
more complicated than doing the same in the polar domain.

Multiplication:
z 1 z 2 =  x 1 + jy 1   x 2 + jy 2  = x 1 x 2 + j  x 1 y 2 + x 2 y 1  + j 2 y 1 y 2
=  x1 x2 – y1 y2  + j  x1 y2 + x2 y1  .

Division:
z1 x 1 + jy 1  x 1 + jy 1  x 2 – jy 2 x1 x2 + j  x2 y1 – x1 y2  – j 2 y1 y2
---- = ------------------ = ------------------ ------------------ = ------------------------------------------------------------------------
z2 x 2 + jy 2 ...